Methods for determining the frequency of periodic signals have many useful applications in a wide range of technologies. As an example, periodic signal frequency analysis is useful in applications where a system produces electrical or mechanical vibration signals. By analyzing these signals and determining the frequencies present in the signal, useful information can be gleaned about the operation of the system. For instance, in applications where electrical motor speed and control are of interest, frequency determination methods are very useful. By analyzing the frequency of the voltage produced by an electric motor, one can determine the speed at which the motor is operating. Knowing this speed, the control system can adjust the electric power supplied to the motor to regulate its speed.
Frequency analysis is also used to determine component failure in machinery. In this application, either electrical signals or mechanical vibrations generated by a machine are measured and compared to the frequencies at which known internal machine parts (i.e. gears) operate is analyzed to determine if these parts are operating at normal operating speeds. Any deviation in this speed will indicate possible fault in that particular part.
Frequency determination is also very important in the field of communications. For example, frequency determinations can be utilized to demodulate a frequency modulated signal. Further, periodic signal determinations are useful where a system is transmitting data at a carrier of unknown frequency. By analyzing the signals produced by the system, one can pinpoint the carrier frequency. The above are just a few examples of the need for accurate and efficient periodic signal analysis.
The determination of the frequency of a periodic signal can be performed by many different methods including both analog and digital methods. For example, one digital method for determining the frequency of a periodic signal involves using a counter which measures the time interval between zero crossings of the periodic signal. By taking the inverse of this time interval, the frequency of the periodic signal is determined. For example, if the time interval between zero crossings is ten seconds, then the frequency of the periodic signal is 0.1 Hz.
Another digital method for the determination of the frequency of a given periodic signal is the zero counting method. This method involves counting the number of times a periodic signal crosses zero during a given time frame. For example, if the signal crosses zero five times during a one second interval the signal has a frequency of 5 Hz.
The frequency of a periodic signal can also be determined from digital analysis of the periodic signal. For example, a periodic signal can be sampled by an analog to digital ("A/D") converter. This A/D converter transforms the analog signal into a set of time-domain digital data points. These digital data points are converted to frequency-domain data points by the use of fourier transforms. As described below, these frequency-domain data points can then be analyzed by frequency determination methods which calculate the frequency of the periodic signal by analyzing the data points.
Digital analysis methods are generally advantageous to analog methods. Analog methods typically require several devices for analyzing the signal, are less accurate, and more susceptible to noise. Digital methods, on the other hand, alleviate these problems by incorporating a digital computer which operates under the control of computer software analyze the digital signal. The use of digital computers and computer software, not only reduces the number of devices needed, it also decreases calculation time and allows for data storage of the signal. Further, digital methods can utilize methods to counteract signal noise and improve accuracy, thereby making digital signal analysis methods more accurate, less susceptible to noise, and flexible. However, digital signal analysis methods do have several limitations. One limitation, in particular, is the limitation on the maximum frequency of a periodic signal that can be properly sampled and determined.
Typically, in digital frequency analysis, an A/D converter obtains digital samples of an analog signal. The A/D typically acquires data samples of the analog signal over a fixed data sampling period. The number of data samples the A/D converter obtains per unit time is known as the sampling rate of the A/D converter. In this manner, the A/D converter produces a discrete number of data points which correspond to the analog signal. These data points are then analyzed with frequency determination methods to calculate the frequency of the periodic signal.
Difficulties occur, however, when the frequency of the analog signal is larger than half the sampling rate of the A/D converter. Half the sampling rate of an A/D converter is called the A/D converter's nyquist frequency. When periodic signals with frequencies exceeding this nyquist frequency are sampled by the A/D converter, the periodic signal is undersampled, and thus the high frequency information of the periodic signal is not uniquely defined, thereby making accurate high signal frequency analysis via conventional signal processing methods impossible.
When periodic signals contain frequencies which exceed the nyquist frequency of the A/D converter, a phenomenon known as aliasing occurs. In this event, frequency signals, known as alias signals, are introduced into the data sampled by the A/D converter. Since the relationship of these alias frequencies to the signal frequency is not uniquely defined, frequency determination by conventional methods is impossible.
Methods have been developed to determine the frequency of periodic signals with frequencies exceeding the nyquist frequency of an A/D converter. These methods utilize the aliasing frequencies that are generated when a periodic signal is undersampled. Knowledge of these aliased signals allows for determination of the frequency of a periodic signal that exceeds the nyquist frequency.
U.S. Pat. No. 5,099,243 to Tsui et al. and U.S. Pat. No. 5,099,194 to Sanderson et al. describe digital methods which use the alias frequencies generated from undersampling of a periodic signal to determine its frequency. Both of these methods determine the frequency of a signal through graphic calculations using the first order aliasing frequencies of two sampled sets of the signal taken at two different sample rates, f.sub.0 and f.sub.1.
Although the methods described by the Tsui et al. and Sanderson et al. determine the frequency of a signal that exceeds the nyquist frequency of an A/D converter, these methods have some drawbacks. First, these two methods require multiple A/D converters for frequency determination. Further, the frequency of the periodic signal can be directly determined only when the two chosen sampling rates, f.sub.0 and f.sub.1, combine in the equation n=f.sub.0 /(f.sub.1 -f.sub.0) such that n is an integer. If this criterion is not met, the frequency can be determined only from the use of look up tables.
Further, both patents disclose that ambiguous solutions are found when two sample rates are used. Sanderson et al. indicates that such ambiguities can be remedied by the use of a third sampling rate. However, this third sampling rate must also satisfy n=f.sub.0 /(f.sub.1 -f.sub.0) such that n is an integer. Further, the Sanderson et al. system requires yet another A/D converter to obtain data at this third sampling rate. Also, both of these methods require calculation of the frequency through graphical methods which are not convenient, practical, or amendable to software implementation.
Finally, neither Tsui et al. nor Sanderson et al. disclose error correction for frequency measurement. Errors such as signal noise and electronic component noise can not only negate the effectiveness of the frequency determination but, in some instances, can produce totally erroneous results. For example, errors due to signal noise are extremely troublesome in frequencies near zero or the nyquist frequency. If the alias frequencies chosen by Sanderson et al. and Tsui et al. to calculate the signal frequency are near zero or the nyquist frequency, these errors will significantly affect the results of the calculation.
Further, and importantly, calculation errors are introduced during the transformation of data samples from time-domain to frequency-domain. In order to increase the accuracy of this transformation, the Tsui et al. and Sanderson et al. methods generally use longer transforms. Unfortunately, these longer transforms increase computing time for frequency determination.
For the foregoing and other reasons, there exists a need for a frequency measurement method for determining the frequency of periodic signals that exceed the nyquist frequency of an A/D converter which reduces the amount of electrical components needed, reduces measurement and calculation errors, and reduces computing and sampling time.